Truth-value semantics

Truth-value semantics

In formal semantics, truth-value semantics is an alternative to Tarskian semantics. It has been primarily championed by Ruth Barcan Marcus, H. Leblanc, and M. Dunn and N. Belnap. It is also called the "substitution interpretation" (of the quantifiers) or substitutional quantification.

The idea of these semantics is that universal (existential) quantifier may be read as a conjunction (disjunction) of formulas in which constants replace the variables in the scope of the quantifier. E.g. ∀xPx may be read (Pa & Pb & Pc &...) where a,b,c are individual constants replacing all occurrences of x in Px.

The main difference between truth-value semantics and the standard semantics for predicate logic is that there are no domains for truth-value semantics. Only the truth clauses for atomic and for quantificational formulas differ from those of the standard semantics. Whereas in standard semantics atomic formulas like Pb or Rca are true if and only if (the referent of) b is a member of the extension of the predicate P, resp., if and only if the pair (c,a) is a member of the extension of R, in truth-value semantics the truth-values of atomic formulas are basic. A universal (existential) formula is true if and only if all (some) substitution instances of it are true. Compare this with the standard semantics which says that a universal (existential) formula is true if and only if for all (some) members of the domain, the formula holds for all (some) of them; e.g. ∀xA is true (under an interpretation) if and only if for all k in the domain D, A(k/x) is true (where A(k/x) is the result of substituting k for all occurrences of x in A). (Here we are assuming that constants are names for themselves--i.e. they are also members of the domain.)

Truth-value semantics is not without its problems. First, the strong completeness theorem and compactness fail. To see this consider the set {F(1), F(2),...}. Clearly the formula ∀xF(x) is a logical consequence of the set, but it is not a consequence of any finite subset of it (and hence it is not deducible from it). It follows immediately that both compactness and the strong completeness theorem fail for truth-value semantics. This is rectified by a modified definition of logical consequence as given in Dunn and Belnap 1968.

Another problem occurs in free logic. Consider a language with one individual constant c that is nondesignating and a predicate F standing for 'does not exist'. Then ∃xFx is false even though a substitution instance (in fact every such instance under this interpretation) of it is true. To solve this problem we simply add the proviso that an existentially quantified statement is true under an interpretation for at least one substitution instance in which the constant designates something that exists.

See also

*Game semantics
*Kripke semantics
*Model-theoretic semantics
*Proof-theoretic semantics
*Truth-conditional semantics

Wikimedia Foundation. 2010.

Игры ⚽ Поможем написать курсовую

Look at other dictionaries:

  • Semantics — is the study of meaning in communication. The word derives from Greek σημαντικός ( semantikos ), significant , [cite web|url= bin/ptext?doc=Perseus%3Atext%3A1999.04.0057%3Aentry%3D%2393797|title=Semantikos, Henry… …   Wikipedia

  • Truth — For other uses, see Truth (disambiguation). Time Saving Truth from Falsehood and Envy, François Lemoyne, 1737 Truth has a variety of meanings, such as the state of being in accord with fact or reality …   Wikipedia

  • Semantics of Business Vocabulary and Business Rules — The Semantics of Business Vocabulary and Business Rules (SBVR) is an adopted standard of the Object Management Group (OMG) intended to be the basis for a formal and detailed natural language declarative description of a complex entity, such as a… …   Wikipedia

  • Formal semantics — See also Formal semantics of programming languages. Formal semantics is the study of the semantics, or interpretations, of formal languages. A formal language can be defined apart from any interpretation of it. This is done by designating a set… …   Wikipedia

  • Cognitive semantics — is part of the cognitive linguistics movement. The main tenets of cognitive semantics are, first, that grammar is conceptualisation; second, that conceptual structure is embodied and motivated by usage; and third, that the ability to use language …   Wikipedia

  • Logical truth — is one of the most fundamental concepts in logic, and there are different theories on its nature. A logical truth is a statement which is true and remains true under all reinterpretations of its components other than its logical constants. It is… …   Wikipedia

  • Degree of truth — In standard mathematics, propositions can typically be considered unambiguously true or false. For instance, the proposition zero belongs to the set { 0 } is regarded as simply true; while the proposition one belongs to the set { 0 } is regarded… …   Wikipedia

  • Extension (semantics) — In any of several studies that treat the use of signs, for example in linguistics, logic, mathematics, semantics, and semiotics, the extension of a concept, idea, or sign consists of the things to which it applies, in contrast with its… …   Wikipedia

  • Proof-theoretic semantics — is an approach to the semantics of logic that attempts to locate the meaning of propositions and logical connectives not in terms of interpretations, as in Tarskian approaches to semantics, but in the role that the proposition or logical… …   Wikipedia

  • situation semantics — An approach to semantics that diverges from the orthodox Fregean tradition in assigning states of affairs to sentences as that to which they refer. The approach needs to avoid the argument known as the slingshot, whereby semanticists following… …   Philosophy dictionary

Share the article and excerpts

Direct link
Do a right-click on the link above
and select “Copy Link”